1. **State the problem:** We have a sample of size $n=25$ from a normal population with mean $\mu=4$ and standard deviation $\sigma=2$. We want to find the probability that the sample mean $\bar{X}$ is less than 5. 2. **Formula and rules:** The sampling distribution of the sample mean $\bar{X}$ is normal with mean $\mu_{\bar{X}} = \mu$ and standard deviation (standard error) $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$. 3. **Calculate the standard error:** $$\sigma_{\bar{X}} = \frac{2}{\sqrt{25}} = \frac{2}{5} = 0.4$$ 4. **Find the z-score for $\bar{X} = 5$:** $$z = \frac{5 - 4}{0.4} = \frac{1}{0.4} = 2.5$$ 5. **Find the probability $P(\bar{X} < 5)$:** This is the same as $P(Z < 2.5)$ for a standard normal distribution. 6. **Use standard normal tables or calculator:** $$P(Z < 2.5) = 0.9938$$ **Final answer:** The probability that the sample mean is less than 5 is **0.9938**.